Penalized $M-$estimators for logistic regression models have been previously study for fixed dimension in order to obtain sparse statistical models and automatic variable selection. In this paper, we derive asymptotic results for penalized $M-$estimators when the dimension $p$ grows to infinity with the sample size $n$. Specifically, we obtain consistency and rates of convergence results, for some choices of the penalty function. Moreover, we prove that these estimators consistently select variables with probability tending to 1 and derive their asymptotic distribution.

翻译：针对逻辑回归模型的惩罚$M-$估计量已在固定维度下得到研究，以实现稀疏统计模型和自动变量选择。本文推导了当维度$p$随样本量$n$趋于无穷大时惩罚$M-$估计量的渐近结果。具体而言，针对某些惩罚函数的选择，我们得到了估计量的一致性和收敛速率结果。此外，我们证明了这些估计量能够以趋于1的概率一致地选择变量，并推导了其渐近分布。